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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 13115234114686--
%%--Suite d\'intégrales--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
Soit $f$ la fonction d\'{e}finie sur $\mathbb{R}_{+}$, par $f(x)=\frac{e^{x}%
}{e^{x}+1}$ et repr\'{e}sent\'{e}e dans le rep\`{e}re de la figure
\ref{fig1_ououe} page \pageref{fig1_ououe}.

\begin{enumerate}
\item D\'{e}terminer une primitive de $f$ sur $\mathbb{R}_{+}$.

\item Soit la suite $(U_{n})$ d\'{e}finie pour $n>0$ par :
\[
U_{n}=\int_{\ln(n)}^{\ln(n+1)}f(x)dx
\]


\begin{enumerate}
\item Calculer $U_{1}$ et $U_{2}$. Exprimer $U_{n}$ en fonction de $n$.

\item Que repr\'{e}sente graphiquement le nombre $U_{n}$ ?
\end{enumerate}

\item Montrer que $U_{n}$ est une suite d\'{e}croissante positive.

Calculer la limite de cette suite.

\item On pose $S_{n}=U_{1}+U_{2}+...+U_{n}$

\begin{enumerate}
\item Calculer $S_{1}$, $S_{2}$, $S_{3}$ et exprimer $S_{n}$ en fonction de
$n$.

\item D\'{e}terminer $\lim_{n\rightarrow+\infty}S_{n}$.%
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\end{enumerate}
\end{enumerate}
\end{exercise}


\end{document}